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To determine if the function \( h(x) \) is concave up or concave down, we need to examine the second differences of the values of \( h(x) \). For a discrete function given in a table, this involves checking the changes in the first differences.

### First Differences
First, we calculate the first differences of \( h(x) \):

| \( x \) | \( h(x) \) | First Difference \( \Delta h(x) \) |
|--------|------------|-------------------------------------|
| 1      | 92         | -                                   |
| 2      | 125        | 125 - 92 = 33                       |
| 3      | 152        | 152 - 125 = 27                      |
| 4      | 173        | 173 - 152 = 21                      |
| 5      | 188        | 188 - 173 = 15                      |
| 6      | 197        | 197 - 188 = 9                       |

### Second Differences
Next, we calculate the second differences of \( h(x) \):

| \( x \) | First Difference \( \Delta h(x) \) | Second Difference \( \Delta^2 h(x) \) |
|--------|-------------------------------------|---------------------------------------|
| 2      | 33                                  | -                                     |
| 3      | 27                                  | 27 - 33 = -6                          |
| 4      | 21                                  | 21 - 27 = -6                          |
| 5      | 15                                  | 15 - 21 = -6                          |
| 6      | 9                                   | 9 - 15 = -6                           |

### Interpretation
The second differences \( \Delta^2 h(x) \) are all \(-6\), which are negative. When the second differences are consistently negative, it indicates that the function \( h(x) \) is concave down.

Therefore, \( h(x) \) is concave down.

Would you like more details or have any questions? Here are some related questions:

1. What does it mean for a function to be concave up?
2. How do you determine concavity from the second derivative in calculus?
3. Can a function change concavity? Provide an example.
4. How is the concept of concavity used in optimization problems?
5. What are some real-life applications of concave up and concave down functions?
6. How can you graphically identify concavity?
7. What is the relationship between concavity and inflection points?
8. How do you find the second differences for non-linear sequences?

**Tip:** For discrete data points, examining the first and second differences can provide insight into the concavity of the function. For continuous functions, the second derivative test is typically used.

Learn how to determine if the function h(x) is concave up or concave down using second differences. This problem involves calculus concepts related to concavity and is suitable for advanced high school students.

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